Further Analysis

# Further Analysis

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$2 . \mathrm{I} . 4 \mathrm{E} \quad$

commentLet $\tau$ be the topology on $\mathbb{N}$ consisting of the empty set and all sets $X \subset \mathbb{N}$ such that $\mathbb{N} \backslash X$ is finite. Let $\sigma$ be the usual topology on $\mathbb{R}$, and let $\rho$ be the topology on $\mathbb{R}$ consisting of the empty set and all sets of the form $(x, \infty)$ for some real $x$.

(i) Prove that all continuous functions $f:(\mathbb{N}, \tau) \rightarrow(\mathbb{R}, \sigma)$ are constant.

(ii) Give an example with proof of a non-constant function $f:(\mathbb{N}, \tau) \rightarrow(\mathbb{R}, \rho)$ that is continuous.

$3 . \mathrm{II} . 17 \mathrm{E} \quad$

comment(i) Explain why the formula

$f(z)=\sum_{n=-\infty}^{\infty} \frac{1}{(z-n)^{2}}$

defines a function that is analytic on the domain $\mathbb{C} \backslash \mathbb{Z}$. [You need not give full details, but should indicate what results are used.]

Show also that $f(z+1)=f(z)$ for every $z$ such that $f(z)$ is defined.

(ii) Write $\log z$ for $\log r+i \theta$ whenever $z=r e^{i \theta}$ with $r>0$ and $-\pi<\theta \leqslant \pi$. Let $g$ be defined by the formula

$g(z)=f\left(\frac{1}{2 \pi i} \log z\right)$

Prove that $g$ is analytic on $\mathbb{C} \backslash\{0,1\}$.

[Hint: What would be the effect of redefining $\log z$ to be $\log r+i \theta$ when $z=r e^{i \theta}$, $r>0$ and $0 \leqslant \theta<2 \pi$ ?]

(iii) Determine the nature of the singularity of $g$ at $z=1$.

2.II.15E

comment(i) Let $X$ be the set of all infinite sequences $\left(\epsilon_{1}, \epsilon_{2}, \ldots\right)$ such that $\epsilon_{i} \in\{0,1\}$ for all $i$. Let $\tau$ be the collection of all subsets $Y \subset X$ such that, for every $\left(\epsilon_{1}, \epsilon_{2}, \ldots\right) \in Y$ there exists $n$ such that $\left(\eta_{1}, \eta_{2}, \ldots\right) \in Y$ whenever $\eta_{1}=\epsilon_{1}, \eta_{2}=\epsilon_{2}, \ldots, \eta_{n}=\epsilon_{n}$. Prove that $\tau$ is a topology on $X$.

(ii) Let a distance $d$ be defined on $X$ by

$d\left(\left(\epsilon_{1}, \epsilon_{2}, \ldots\right),\left(\eta_{1}, \eta_{2}, \ldots\right)\right)=\sum_{n=1}^{\infty} 2^{-n}\left|\epsilon_{n}-\eta_{n}\right|$

Prove that $d$ is a metric and that the topology arising from $d$ is the same as $\tau$.

3.I.5E

commentLet $C$ be the contour that goes once round the boundary of the square

$\{z:-1 \leqslant \operatorname{Re} z \leqslant 1,-1 \leqslant \operatorname{Im} z \leqslant 1\}$

in an anticlockwise direction. What is $\int_{C} \frac{d z}{z}$ ? Briefly justify your answer.

Explain why the integrals along each of the four edges of the square are equal.

Deduce that $\int_{-1}^{1} \frac{d t}{1+t^{2}}=\frac{\pi}{2}$.

4.I.4E

comment(i) Let $D$ be the open unit disc of radius 1 about the point $3+3 i$. Prove that there is an analytic function $f: D \rightarrow \mathbb{C}$ such that $f(z)^{2}=z$ for every $z \in D$.

(ii) Let $D^{\prime}=\mathbb{C} \backslash\{z \in \mathbb{C}: \operatorname{Im} z=0$, Re $z \leqslant 0\}$. Explain briefly why there is at most one extension of $f$ to a function that is analytic on $D^{\prime}$.

(iii) Deduce that $f$ cannot be extended to an analytic function on $\mathbb{C} \backslash\{0\}$.

4.II.14E

comment(i) State and prove Rouché's theorem.

[You may assume the principle of the argument.]

(ii) Let $0<c<1$. Prove that the polynomial $p(z)=z^{3}+i c z+8$ has three roots with modulus less than 3. Prove that one root $\alpha$ satisfies $\operatorname{Re} \alpha>0, \operatorname{Im} \alpha>0$; another, $\beta$, satisfies $\operatorname{Re} \beta>0$, Im $\beta<0$; and the third, $\gamma$, has $\operatorname{Re} \gamma<0$.

(iii) For sufficiently small $c$, prove that $\operatorname{Im} \gamma>0$.

[You may use results from the course if you state them precisely.]

$2 . \mathrm{I} . 4 \mathrm{E} \quad$

commentLet $\tau_{1}$ be the collection of all subsets $A \subset \mathbb{N}$ such that $A=\emptyset$ or $\mathbb{N} \backslash A$ is finite. Let $\tau_{2}$ be the collection of all subsets of $\mathbb{N}$ of the form $I_{n}=\{n, n+1, n+2, \ldots\}$, together with the empty set. Prove that $\tau_{1}$ and $\tau_{2}$ are both topologies on $\mathbb{N}$.

Show that a function $f$ from the topological space $\left(\mathbb{N}, \tau_{1}\right)$ to the topological space $\left(\mathbb{N}, \tau_{2}\right)$ is continuous if and only if one of the following alternatives holds:

(i) $f(n) \rightarrow \infty$ as $n \rightarrow \infty$;

(ii) there exists $N \in \mathbb{N}$ such that $f(n)=N$ for all but finitely many $n$ and $f(n) \leqslant N$ for all $n$.

$3 . \mathrm{I} . 3 \mathrm{E} \quad$

comment(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $|f(z)| \leqslant 1+|z|^{1 / 2}$ for every $z$. Prove that $f$ is constant.

(b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $\operatorname{Re}(f(z)) \geqslant 0$ for every $z$. Prove that $f$ is constant.

2.II.13E

comment(a) Let $f:[1, \infty) \rightarrow \mathbb{C}$ be defined by $f(t)=t^{-1} e^{2 \pi i t}$ and let $X$ be the image of $f$. Prove that $X \cup\{0\}$ is compact and path-connected. [Hint: you may find it helpful to set $\left.s=t^{-1} .\right]$

(b) Let $g:[1, \infty) \rightarrow \mathbb{C}$ be defined by $g(t)=\left(1+t^{-1}\right) e^{2 \pi i t}$, let $Y$ be the image of $g$ and let $\bar{D}$ be the closed unit $\operatorname{disc}\{z \in \mathbb{C}:|z| \leq 1\}$. Prove that $Y \cup \bar{D}$ is connected. Explain briefly why it is not path-connected.

3.II.13E

comment(a) State Taylor's Theorem.

(b) Let $f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}$ and $g(z)=\sum_{n=0}^{\infty} b_{n}\left(z-z_{0}\right)^{n}$ be defined whenever $\left|z-z_{0}\right|<r$. Suppose that $z_{k} \rightarrow z_{0}$ as $k \rightarrow \infty$, that no $z_{k}$ equals $z_{0}$ and that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $a_{n}=b_{n}$ for every $n \geqslant 0$.

(c) Let $D$ be a domain, let $z_{0} \in D$ and let $\left(z_{k}\right)$ be a sequence of points in $D$ that converges to $z_{0}$, but such that no $z_{k}$ equals $z_{0}$. Let $f: D \rightarrow \mathbb{C}$ and $g: D \rightarrow \mathbb{C}$ be analytic functions such that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $f(z)=g(z)$ for every $z \in D$.

(d) Let $D$ be the domain $\mathbb{C} \backslash\{0\}$. Give an example of an analytic function $f: D \rightarrow \mathbb{C}$ such that $f\left(n^{-1}\right)=0$ for every positive integer $n$ but $f$ is not identically 0 .

(e) Show that any function with the property described in (d) must have an essential singularity at the origin.

4.I.4E

comment(a) State and prove Morera's Theorem.

(b) Let $D$ be a domain and for each $n \in \mathbb{N}$ let $f_{n}: D \rightarrow \mathbb{C}$ be an analytic function. Suppose that $f: D \rightarrow \mathbb{C}$ is another function and that $f_{n} \rightarrow f$ uniformly on $D$. Prove that $f$ is analytic.

4.II.13E

comment(a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.

(b) Let $p(z)=z^{5}+z$. Find all $z$ such that $|z|=1$ and $\operatorname{Im}(p(z))=0$. Calculate $\operatorname{Re}(p(z))$ for each such $z$. [It will be helpful to set $z=e^{i \theta}$. You may use the addition formulae $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$ and $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$.]

(c) Let $\gamma:[0,2 \pi] \rightarrow \mathbb{C}$ be the closed path $\theta \mapsto e^{i \theta}$. Use your answer to (b) to give a rough sketch of the path $p \circ \gamma$, paying particular attention to where it crosses the real axis.

(d) Hence, or otherwise, determine for every real $t$ the number of $z$ (counted with multiplicity) such that $|z|<1$ and $p(z)=t$. (You need not give rigorous justifications for your calculations.)

2.I.4G

commentLet the function $f=u+i v$ be analytic in the complex plane $\mathbb{C}$ with $u, v$ real-valued.

Prove that, if $u v$ is bounded above everywhere on $\mathbb{C}$, then $f$ is constant.

2.II.13G

comment(a) Given a topology $\mathcal{T}$ on $X$, a collection $\mathcal{B} \subseteq \mathcal{T}$ is called a basis for $\mathcal{T}$ if every non-empty set in $\mathcal{T}$ is a union of sets in $\mathcal{B}$. Prove that a collection $\mathcal{B}$ is a basis for some topology if it satisfies:

(i) the union of all sets in $\mathcal{B}$ is $X$;

(ii) if $x \in B_{1} \cap B_{2}$ for two sets $B_{1}$ and $B_{2}$ in $\mathcal{B}$, then there is a set $B \in \mathcal{B}$ with $x \in B \subset B_{1} \cap B_{2}$.

(b) On $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ consider the dictionary order given by

$\left(a_{1}, b_{1}\right)<\left(a_{2}, b_{2}\right)$

if $a_{1}<a_{2}$ or if $a_{1}=a_{2}$ and $b_{1}<b_{2}$. Given points $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ let

$\langle\mathbf{x}, \mathbf{y}\rangle=\left\{\mathbf{z} \in \mathbb{R}^{2}: \mathbf{x}<\mathbf{z}<\mathbf{y}\right\}$

Show that the sets $\langle\mathbf{x}, \mathbf{y}\rangle$ for $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ form a basis of a topology.

(c) Show that this topology on $\mathbb{R}^{2}$ does not have a countable basis.

3.I.3G

commentLet $f: X \rightarrow Y$ be a continuous map between topological spaces. Let

$G_{f}=\{(x, f(x)): x \in X\} .$

(a) Show that if $Y$ is Hausdorff, then $G_{f}$ is closed in $X \times Y$.

(b) Show that if $X$ is compact, then $G_{f}$ is also compact.

3.II.13G

comment(a) Let $f$ and $g$ be two analytic functions on a domain $D$ and let $\gamma \subset D$ be a simple closed curve homotopic in $D$ to a point. If $|g(z)|<|f(z)|$ for every $z$ in $\gamma$, prove that $\gamma$ encloses the same number of zeros of $f$ as of $f+g$.

(b) Let $g$ be an analytic function on the disk $|z|<1+\epsilon$, for some $\epsilon>0$. Suppose that $g$ maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that $g$ has exactly one fixed point in the open unit disk.

(c) Prove that, if $|a|<1$, then

$z^{m}\left(\frac{z-a}{1-\bar{a} z}\right)^{n}-a$

has $m+n$ zeros in $|z|<1$.

4.I.4G

comment(a) Let $X$ be a topological space and suppose $X=C \cup D$, where $C$ and $D$ are disjoint nonempty open subsets of $X$. Show that, if $Y$ is a connected subset of $X$, then $Y$ is entirely contained in either $C$ or $D$.

(b) Let $X$ be a topological space and let $\left\{A_{n}\right\}$ be a sequence of connected subsets of $X$ such that $A_{n} \cap A_{n+1} \neq \emptyset$, for $n=1,2,3, \ldots$. Show that $\bigcup_{n \geqslant 1} A_{n}$ is connected.

4.II.13G

commentA function $f$ is said to be analytic at $\infty$ if there exists a real number $r>0$ such that $f$ is analytic for $|z|>r$ and $\lim _{z \rightarrow 0} f(1 / z)$ is finite (i.e. $f(1 / z)$ has a removable singularity at $z=0)$. $f$ is said to have a pole at $\infty$ if $f(1 / z)$ has a pole at $z=0$. Suppose that $f$ is a meromorphic function on the extended plane $\mathbb{C}_{\infty}$, that is, $f$ is analytic at each point of $\mathbb{C}_{\infty}$ except for poles.

(a) Show that if $f$ has a pole at $z=\infty$, then there exists $r>0$ such that $f(z)$ has no poles for $r<|z|<\infty$.

(b) Show that the number of poles of $f$ is finite.

(c) By considering the Laurent expansions around the poles show that $f$ is in fact a rational function, i.e. of the form $p / q$, where $p$ and $q$ are polynomials.

(d) Deduce that the only bijective meromorphic maps of $\mathbb{C}_{\infty}$ onto itself are the Möbius maps.

2.I.4B

commentDefine the terms connected and path connected for a topological space. If a topological space $X$ is path connected, prove that it is connected.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} I=\{(x, 0): 0 \leq x \leq 1\}, \quad A=\left\{(0, y): \frac{1}{2} \leq y \leq 1\right\}, \text { and } \\ J_{n}=\left\{\left(n^{-1}, y\right): 0 \leq y \leq 1\right\} \quad \text { for } n \geq 1 \end{gathered}$

Let

$X=A \cup I \cup \bigcup_{n \geq 1} J_{n}$

with the subspace (metric) topology. Prove that $X$ is connected.

[You may assume that any interval in $\mathbb{R}$ (with the usual topology) is connected.]

2.II.13A

commentState Liouville's Theorem. Prove it by considering

$\int_{|z|=R} \frac{f(z) d z}{(z-a)(z-b)}$

and letting $R \rightarrow \infty$.

Prove that, if $g(z)$ is a function analytic on all of $\mathbb{C}$ with real and imaginary parts $u(z)$ and $v(z)$, then either of the conditions:

$\text { (i) } u+v \geqslant 0 \text { for all } z \text {; or (ii) } u v \geqslant 0 \text { for all } z \text {, }$

implies that $g(z)$ is constant.

3.I.3B

commentState a version of Rouché's Theorem. Find the number of solutions (counted with multiplicity) of the equation

$z^{4}=a(z-1)\left(z^{2}-1\right)+\frac{1}{2}$

inside the open disc $\{z:|z|<\sqrt{2}\}$, for the cases $a=\frac{1}{3}, 12$ and 5 .

[Hint: For the case $a=5$, you may find it helpful to consider the function $\left(z^{2}-1\right)(z-$ 2) $(z-3)$.]

3.II.13B

commentIf $X$ and $Y$ are topological spaces, describe the open sets in the product topology on $X \times Y$. If the topologies on $X$ and $Y$ are induced from metrics, prove that the same is true for the product.

What does it mean to say that a topological space is compact? If the topologies on $X$ and $Y$ are compact, prove that the same is true for the product.

4.I.4A

commentLet $f(z)$ be analytic in the $\operatorname{disc}|z|<R$. Assume the formula

$f^{\prime}\left(z_{0}\right)=\frac{1}{2 \pi i} \int_{|z|=r} \frac{f(z) d z}{\left(z-z_{0}\right)^{2}}, \quad 0 \leqslant\left|z_{0}\right|<r<R .$

By combining this formula with a complex conjugate version of Cauchy's Theorem, namely

$0=\int_{|z|=r} \overline{f(z)} d \bar{z},$

prove that

$f^{\prime}(0)=\frac{1}{\pi r} \int_{0}^{2 \pi} u(\theta) e^{-i \theta} d \theta$

where $u(\theta)$ is the real part of $f\left(r e^{i \theta}\right)$.

4.II.13B

commentLet $\Delta^{*}=\{z: 0<|z|<r\}$ be a punctured disc, and $f$ an analytic function on $\Delta^{*}$. What does it mean to say that $f$ has the origin as (i) a removable singularity, (ii) a pole, and (iii) an essential singularity? State criteria for (i), (ii), (iii) to occur, in terms of the Laurent series for $f$ at 0 .

Suppose now that the origin is an essential singularity for $f$. Given any $w \in \mathbb{C}$, show that there exists a sequence $\left(z_{n}\right)$ of points in $\Delta^{*}$ such that $z_{n} \rightarrow 0$ and $f\left(z_{n}\right) \rightarrow w$. [You may assume the fact that an isolated singularity is removable if the function is bounded in some open neighbourhood of the singularity.]

State the Open Mapping Theorem. Prove that if $f$ is analytic and injective on $\Delta^{*}$, then the origin cannot be an essential singularity. By applying this to the function $g(1 / z)$, or otherwise, deduce that if $g$ is an injective analytic function on $\mathbb{C}$, then $g$ is linear of the form $a z+b$, for some non-zero complex number $a$. [Here, you may assume that $g$ injective implies that its derivative $g^{\prime}$ is nowhere vanishing.]

Part IB